Blending Functions Research Articles

Capturing transition around low-Reynolds number hydrofoil with zero-equation transition model

Compared with the local correlation-based shear stress transport (SST) γ−Reθ transition model (where SST k−ω transport equations are coupled with intermittency γ and transitional momentum-thickness Reynolds number Reθ transport equations), relatively simple and convenient modifications are applied to the parent SST k−ω model for computing natural and separation-induced transitions over the hydrofoil at a low-Reynolds number (LRN). The curiosity toward hydrofoil performance at an LRN has been enhanced by increasing attention to autonomous marine systems, deserving numerical simulations for transitional flow using computational fluid dynamics. With the newly devised transitional SST (T-SST) model, the viscous sublayer blending function F2 is slightly modified, and a stress-intensity parameter as a function of eddy-to-laminar viscosity ratio RT is introduced; intended formulations are plausible and have significant impacts on the transition prediction. Owing to the inherent potential for predicting bypass transition, two anisotropic versions of the v¯2−f(V2F) turbulence model are selected to evaluate their competencies in capturing separation-induced and natural transitions. Results demonstrate that natural transition prediction is more challenging than separation-induced transition for the V2F model. Nonetheless, the T-SST model performs consistently well in replicating both transitional phenomena.

An equivalent parameter geometric shape representation using independent coordinates of cubic Bézier control points

Bézier surface has been commonly applied to represent the complex geometric shape. Generally, all control points are dealt with by the same blending functions, regardless of the effect of independent coordinate. It causes to lack the modeling flexibility. Therefore, this paper proposes an equivalent parameter geometric shape representation method using the independent coordinates of control points. Since the coordinate components of control points are independent, the geometric modeling becomes more flexible. Firstly, a general Bézier curve is described in detail. Related expression is brought out in the form of independent coordinates by introducing two parameters. Then, their geometric meanings are analyzed in detail. Since both parameters are independent to parametric variables u and v, Bézier curve possess the same interval in the discrete parametric space, namely equivalent parameter. Next, a bicubic Bézier subsurface patch representation is discussed, including regular and non-regular subsurface patch. A general surface expression is given out in the form of independent coordinates, as well as the parameter structure and the geometric transformations. Finally, an example of ‘Bézier tree branch’ is constructed by using the proposed method. Results shows that the proposed method is feasible and reasonable.

A computational framework combining a basic turbulent combustion model, self-similar chemistry behavior, and a wall thermal law adapted for large-eddy simulations of a cyclically operating constant volume combustion chamber

The development of a computational tool capable of simulating and completing the experimental data obtained on a constant volume cyclic combustion (CV2) bench available at the Pprime Institute is the major objective of this study. The main difficulty is that, no matter how good the models used or the mesh refinement, specifying the boundary and initial conditions for the simulation of a specific cycle independently of the previous cycles requires an unattainable amount and accuracy of experimental data. In addition, high-fidelity simulation of all previous cycles is too costly. Developing a modeling framework compatible with coarse meshes for which the computational cost is reasonable is the first step necessary to simulate multiple cycles. However, the LES of CV2 presents numerous modeling difficulties due to the interactions of turbulent flow with fast and slow chemical reactions, compressible effects, walls and heat losses. Consequently, the wide spectrum of unresolved scales in the presence of so many correlated physical mechanisms leads to the introduction of several modeling parameters. Specific modeling developments are provided in this paper for turbulent combustion in the presence of residual burnt gases and the database takes into account the self-similar nature of the thermochemical properties. A detailed time-function for the inlet and outlet mass flow rates and a new blending function for the wall heat fluxes are also introduced. The sensitivity of the numerical results to the values of key modeling parameters such as injected air mass mair, ejected mass mout, segregation rate S∈[0.7;0.8], wall heat losses parameter κT∈[0.41;2.36] is studied. A quantitative comparison of the results and computational data is performed based on the pressure and heat flux signals in the case of propane-air combustion. The chamber pressure and temperature of air is 1 bar and 307K for the initial cycle. The calculated pressure profiles are in very good agreement with the experimental results. The values obtained for the calculated pressure peaks respectively for the first three reactive cycles are 17.4, 13.1 and 14.4 bars. For the experimental pressure peaks, the values obtained are 17.8, 14 and 14.6 bars which shows the validity of the modeling strategy. A qualitative comparison of the numerical results with other measurements, i.e PIV, flame visualization and MIE tomography, is also presented. The numerical results are then used to describe the flow during the successive phases, i.e injection, combustion and exhaust, of one specific cycle. Finally, the numerical pressure signal and wall heat flux obtained from the simulation of a sequence of ten cycles is compared to the experimental data. This last result demonstrates the ability of the LES tool to study the cycle-to-cycle variability observed in CV2 but further work is now required.

An Application of Variational Minimization: Quasi-Harmonic Coons Patches

For a minimal surface, the mean curvature of the surface vanishes for all possible parameterizations which results in a second-order nonlinear partial differential equation ( p d e ), whose solution in general is the desired surface as the unknown function of surface parameters. The solution of this partial differential equation is known only for very few cases. Instead of solving the corresponding partial differential equation, we exploit an ansatz method (Ahmad et al. (2013), Ahmad et al. (2014) Ahmad et al. (2015)), used for Coons patches spanned by finite number of boundary curves for a quasi-minimal surface as the extremal of r m s of mean curvature. The ansatz method targets a slightly perturbed surface (the rational blending interpolants-based Coons patch, Hermite cubic polynomials interpolants-based Coons patch, and Ferguson surface (vanishing twist vectors) in our case) that comprises initially a nonminimal surface plus the product of a real parameter with a variational function of surface parameters (vanishes at the boundary curves along the unit normal to the slightly perturbed surface). The variational function can be deliberately chosen as the product of linear functions and the mean curvature of the initial nonminimal surface such that it is zero at the boundary curves and then replace this mean curvature by the mean curvature of the resulting surface to find a surface of reduced area, and the process can be repeated for further improvement. In this article, we extend the ansatz method (1) for the rational blending functions interpolants-based Coons patch with a parameter in their form for its different values and (2) for the blending functions comprising Hermite cubic polynomial interpolants-based Coons patch (bicubically blended Coons patch (BBCP) and the Ferguson surface). The ansatz method can be extended for the variational extremal of the surfaces for fuzzy optimal control problems (Filev et al. (1992), Emamizadeh (2005), Farhadinia (2011), Mustafa et al. (2021)).

An Accurate State Visualization of Multiplexed and PWM Fed Peripherals in the Virtual Simulators of Embedded Systems

A method of precise and real-like embedded systems peripherals visualization in virtual simulations is presented in the article. For high frequency peripheral inputs, blended graphics of real-look representations of peripherals are used, allowing for an accurate visualization with consideration of their input signal nature based on PWM and multiplexing schemes. An accurate simulator of a didactic embedded system called MKD-51 is introduced as a plug in for Keil μVision IDE. A comparison with a real microprocessor system using multiplexing seven-segment LED modules as well as LED bar fed by PWM is shown. The achieved quantity results are included to illustrate compliance with real devices and improvements in visualization accuracy due to applying additional blending mapping function. Brightness estimation differences of LED visualization compared to a real device are at a single percentage level for seven-segment multiplexed modules.

Concurrent multiscale analysis without meshing: Microscale representation with CutFEM and micro/macro model blending

In this paper, we develop a novel unfitted multiscale framework that combines two separate scales represented by only one single computational mesh. Our framework relies on a mixed zooming technique where we zoom at regions of interest to capture microscale properties and then mix the micro and macroscale properties in a transition region. Furthermore, we use homogenization techniques to derive macro model material properties. The microscale features are discretized using CutFEM. The transition region between the micro and macroscale is represented by a smooth blending function. To address the issues with ill-conditioning of the multiscale system matrix due to the arbitrary intersections in cut elements and the transition region, we add stabilization terms acting on the jumps of the normal gradient (ghost-penalty stabilization). We show that our multiscale framework is stable and is capable to reproduce mechanical responses for heterogeneous structures in a mesh-independent manner. The efficiency of our methodology is exemplified by 2D and 3D numerical simulations of linear elasticity problems.

A model for Real Estate Price Prediction using Multi-Level Stacking Ensemble Technique

Recent research and economic publications have shown the impact of real estate investment on the over economy of Nigeria. It is therefore crucial to employ machine learning technique to predict the price for real estate properties. Real estate price analysis and prediction will assist in establishment of real estate policies and can also be used to aid real estate property stakeholders to come up with informative decisions without bias or prejudice. Thus, it is imperative to develop a model to improve the accuracy of real estate price prediction. The goal of this research is to develop a model using a multi-level stacking ensemble technique to predict price of real estate property. The dataset utilized for the study was collected from transactions done by real estate firms in Port Harcourt and it consist of a total of 1053 rows with twelve features. The base model used includes Random Forest(RF), Extreme Gradient Boosting Algorithm(XGBoost), Light Gradient Boosting Machine(LightGBM), Decision Tree regression and ElasticNet Regression. Various combinations of the base models were stacked using StackingCVRegressor. The final model was developed by combining the best performing stacked models and evaluated using R-Square, Mean Absolute Error(MAE), Root Mean Square Error(RMSE), Mean Square Error(MSE) and Training time. The proposed model outperformed the various individual base model with R-square of 0.985203, MSE of 0.013438, RMSE of 0.115923, MAE of 0.063411 and training time of 0.599398. The result show that multi-level stacking significant improve the accuracy of a model. Again, it was observed stacking improve the performance accuracy of a model at the cost of computational time. Stacking by using blending function for the proposed model significantly reduced the computational time for training the model to 0.599398 second when compared to using StackingCVRegressor with training time of 107.054931 seconds. Therefore, multi-level stacking ensemble technique can be employed to improve the predictive accuracy of a prediction model. Future work can be done by increasing the dataset and also increasing the number of features.

Non-Rational and Rational Transfinite Interpolation Using Bernstein Polynomials

It is shown that the standard transfinite interpolation in quadrilateral patches may be written in terms of two sets of Bernstein polynomials. The former set is of the same degree with the corresponding blending functions while the latter is of the same degree with the Lagrange polynomials which operate like trial functions along each of the four edges as well as the additional inter-boundaries of the patch. The replacement of the Lagrange polynomials by the Bernstein ones allows the use of control points useful for design. Also it allows the determination of weights that may ensure accurate representation of quadric surfaces including those of revolution (spheres, ellipsoids, hyperboloids, etc). The presentation restricts to the standard Gordon formulation, which refers to structured stencils, similar to rectangular plates reinforced with longitudinal and transverse stiffeners. As an example, the proposed formula is applied to the geometric representation of a cylindrical and a spherical patch. In the latter case a nonlinear programming optimization technique was applied, starting with initial data pertinent to a tensor product surface, from which original weights and control points were obtained for the accurate representation of a spherical cap. In addition, the involved shape functions were successfully applied to the numerical solution of a boundary value problem.

Generalized B\'{e}zier curves based on Bernstein-Stancu-Chlodowsky type operators

In this paper, we use the blending functions of Bernstein-Stancu-Chlodowsky type operators with shifted knots for construction of modified Chlodowsky B\'{e}zier curves. We study the nature of degree elevation and degree reduction for B\'{e}zier Bernstein-Stancu-Chlodowsky functions with shifted knots for $t \in [\frac{\gamma}{n+\delta},\frac{n+\gamma}{n+\delta}]$. We also present a de Casteljau algorithm to compute Bernstein B\'{e}zier curves with shifted knots. The new curves have some properties similar to B\'{e}zier curves. Furthermore, some fundamental properties for Bernstein B\'{e}zier curves are discussed. Our generalizations show more flexibility in taking the value of $\gamma$ and $\delta$ and advantage in shape control of curves. The shape parameters give more convenience for the curve modelling.

Membrane fiber element for reinforced concrete walls – the benefits of macro and micro modeling approaches

A novel element that mixes the formulation and versatility of microscopic models, and the mesh refinement of macroscopic models, was developed to predict the nonlinear response of reinforced concrete walls. The element, named as Membrane Fiber Element (MEFI), is described by four nodes, each containing three degrees of freedom (DOFs), two translations, and one in-plane rotation (drilling) DOF, which incorporates a blended interpolation function for the displacements over the element. The element formulation accommodates the quadrature points and weights of the classical finite element formulation of membrane elements to resemble strips (fibers), similarly to macroscopic elements. In this modeling approach, a RC panel element that follows a fixed-angle approach was used. The element was validated against five RC wall specimen tests with rectangular cross-sections and different aspect ratios reported in the literature. Also, the element was validated for walls with discontinuities using data from an experimental program of walls with openings. Comparisons of model predictions and experimental results, reveal that the MEFI element is reliable to predict the nonlinear response of reinforced concrete walls with different aspect ratios and properties, requiring less mesh refinement, and therefore less running time, for an accurate response estimation compared to finite element models.

Lupaş post quantum blending functions and Bézier curves over arbitrary intervals

In this paper, we extend the properties of rational Lupa?-Bernstein blending functions, Lupa?-B?zier curves and surfaces over arbitrary compact intervals [?,?] in the frame of post quantum-calculus and derive the de-Casteljau?s algorithm based on post quantum-integers. We construct a two parameter family as Lupa? post quantum Bernstein functions over arbitrary compact intervals and establish their degree elevation and reduction properties. We also discuss some fundamental properties over arbitrary intervals for these curves such as de Casteljau algorithm and degree evaluation properties. Further we construct post quantum Lupa? Bernstein operators over arbitrary compact intervals with the help of rational Lupa?- Bernstein functions. At the end some graphical representations are added to demonstrate consistency of theoretical findings.

Differentiable Direct Volume Rendering.

We present a differentiable volume rendering solution that provides differentiability of all continuous parameters of the volume rendering process. This differentiable renderer is used to steer the parameters towards a setting with an optimal solution of a problem-specific objective function. We have tailored the approach to volume rendering by enforcing a constant memory footprint via analytic inversion of the blending functions. This makes it independent of the number of sampling steps through the volume and facilitates the consideration of small-scale changes. The approach forms the basis for automatic optimizations regarding external parameters of the rendering process and the volumetric density field itself. We demonstrate its use for automatic viewpoint selection using differentiable entropy as objective, and for optimizing a transfer function from rendered images of a given volume. Optimization of per-voxel densities is addressed in two different ways: First, we mimic inverse tomography and optimize a 3D density field from images using an absorption model. This simplification enables comparisons with algebraic reconstruction techniques and state-of-the-art differentiable path tracers. Second, we introduce a novel approach for tomographic reconstruction from images using an emission-absorption model with post-shading via an arbitrary transfer function.

Layer-wise optimization of elliptical laminates with curvilinear fibers

The lay-ups of symmetric variable stiffness composite laminated elliptical plates with curvilinear fibers are optimized for the maximum frequencies of the lowest three modes using the layer-wise optimization method. Two design variables and three constraints are considered. The fiber paths are constructed using the method of shifted paths. The laminate is modeled as one curved hierarchical finite element. The formulation is based on the first-order shear deformation theory. The blending function method is used to model accurately the geometry. New results for the optimal lay-ups that maximize the natural frequencies of the lowest three modes are obtained. The method is validated based on convergence test and comparison with published results for symmetric constant stiffness composite laminated elliptical plates with rectilinear fibers. A parametric study is performed showing the effects of aspect ratio, thickness ratio, modulus ratio, number of layers, and boundary constraint on the optimal solutions.

Application of Hybrid Reynolds Averaged Navier–Stokes and Large-Eddy-Simulation Eddy Viscosity Blending Methods in OVERFLOW

A myriad of different blending or switching functions have been developed to blend Reynolds-averaged Navier–Stokes (RANS) and large-eddy-simulation (LES) models properly in order to prevent issues such as modeled stress depletion. This challenge is a large focus of modern hybrid methods along with generating sufficient turbulent statistics in the Reynolds-averaged to large-eddy-simulation interface. This paper investigates these issues by implementing a variety of blending functions in the NASA OVERFLOW computational fluid dynamics solver and comparing them with the existing length-scale blending models. Scenarios considered include a zero pressure gradient flat plate, subcritical flow around a cylinder, and a supersonic reattaching shear layer. Overall, the eddy-viscosity blending methods produce results similar, if not improved, to their length-scale blending counterparts. In particular, it was found that the entropy function gave more control and stability during the transition from RANS to LES.

Error estimation of a homogenized streamwise periodic boundary layer

This work concerns the simulation of a homogenized streamwise periodic boundary layer which reduces the computational cost of direct numerical simulations of incompressible flat plate turbulent boundary layers. It expands upon the understanding of how the transpiration velocity changes dependent on whether a blending function is used to account for inner and outer self-similar scaling effects. With this rescaling correction, lower Reynolds number effects on the transpiration velocity are captured by the homogenized streamwise periodic boundary layer simulation.

Three-Dimensional Solution for the Vibration Analysis of Functionally Graded Rectangular Plate with/without Cutouts Subject to General Boundary Conditions.

Functionally graded materials (FGMs) structures are increasingly used in engineering due to their superior mechanical and material properties, and the FGMs plate with cutouts is a common structural form, but research on the vibration characteristics of FGMs plate with cutouts is relatively limited. In this paper, the three-dimensional exact solution for the vibration analysis of FGMs rectangular plate with circular cutouts subjected to general boundary conditions is presented based on the three-dimensional elasticity theory. The displacement field functions are expressed as standard cosine Fourier series plus auxiliary cosine series terms satisfying the boundary conditions in the global coordinate system. The plate with circular cutout is discretized into four curve quadrilateral sub-domains using the p-version method, and then the blending function method is applied to map the closed quadrilateral region to the computational space. The characteristic equation is obtained based on the Lagrangian energy principle and Rayleigh–Ritz method. The efficiency and reliability of proposed method are verified by comparing the present results with those available in the literature and FEM methods. Finally, a parametric study is investigated including the cutout sizes, the cutout positions, and the cutout numbers from the free vibration characteristic analysis and the harmonic analysis. The results can serve as benchmark data for other research on the vibration of FGMs plates with cutouts.

Development of a semi‐implicit contact methodology for finite volume stress solvers

AbstractThe past decades have seen numerous efforts to apply the finite volume methodology to solid mechanics problems. However, only limited work has been done by the finite volume community toward the simulation of mechanical contact. In this article, we present a novel semi‐implicit methodology for the solution of static force‐loading contact problems with cell‐centered finite volume codes. Starting from the similarities with multi‐material problems, we derive an implicit discretization scheme for the normal contact stress with a straightforward inclusion of frictional forces and correction vectors for non‐orthogonal boundaries. With the introduction of a sigmoid blending function interpolating between contact stresses and gap pressure, the proposed approach is extended to cases with partially closed gap. The contact procedure is designed around an arbitrary mesh mapping algorithm to allow for non‐conformal meshes at the contact interface between the two bodies. Finally, we verify the contact methodology against five benchmarks cases.

Three-dimensional free bending vibrations of variable radius functionally graded circular column immersed in infinite fluid

The three-dimensional free bending vibrations of variable radius functionally graded circular column fully or partially immersed in infinite fluid are investigated for the first time using a coupled hierarchical finite-infinite element method based on curved elements. The fluid is incompressible, irrotational, and inviscid. The material properties vary continuously along the radial direction according to a simple power law function. The solid and fluid behaviors are described by the three-dimensional elasticity theory and Laplace’s equation, respectively. Coupling occurs on the interface via the boundary restraints imposed on it. Curved edges are described exactly by the blending functions. Examples of clamped-free and clamped-guided functionally graded circular columns with quadratic and cubic radius variations are used to illustrate the method. Convergence test and comparison study are conducted. New results for the frequencies of the lowest two bending modes are provided, which may serve as benchmark solutions. Furthermore, a parametric study is conducted to show the effects of various geometrical and mechanical parameters on the frequencies, modal deflections, and modal hydrodynamic pressures.

Predicting vapour transport from semi-volatile organic compounds concealed within permeable packaging

The vapour concentration present in enclosed spaces containing concealed semi-volatile organic compounds (SVOCs), such as explosives, is difficult to measure experimentally. Therefore, mathematical models play a key role in understanding the transport of these materials. Vapour transport has previously been modelled in a range of environments, from small emission cells to whole rooms, using both analytical and numerical approaches. These models typically include either a well-mixed air volume or a simple sorption model. This work has been extended by including a multi-layer vapour sorption/permeation model within a computational fluid dynamics (CFD) framework. This allows for vapour source terms from items concealed within permeable packaging to be considered. The CFD based permeation model includes sorption/desorption, using a linear isotherm at inner and outer surfaces and a blended wall function to account for the effects of near-wall turbulence. The model has been validated for the explosive SVOC, ethylene glycol dinitrate (EGDN). The model has been used to show how vapour concentrations around a cardboard box containing a SVOC vary when some of the key input parameters are changed. Changing the vapour source from EGDN to the much lower vapour pressure trinitrotoluene (TNT), had a significant effect, as expected, and this was most pronounced early on due to the difference in permeation lag times for the two materials. Conversely, changing the type of cardboard had only a small effect on the concentrations. This type of modelling approach can now be used to study a wide range of SVOC transport problems which would not previously have been possible.

Lattice Boltzmann method for fluid–structure interaction in compressible flow

We present a two-way coupled fluid–structure interaction scheme for rigid bodies using a two-population lattice Boltzmann formulation for compressible flows. An arbitrary Lagrangian–Eulerian formulation of the discrete Boltzmann equation on body-fitted meshes is used in combination with polynomial blending functions. The blending function approach localizes mesh deformation and allows treating multiple moving bodies with a minimal computational overhead. We validate the model with several test cases of vortex induced vibrations of single and tandem cylinders and show that it can accurately describe dynamic behavior of these systems. Finally, in the compressible regime, we demonstrate that the proposed model accurately captures complex phenomena such as transonic flutter over an airfoil.